There is a common rule of thumb that I have often seen presented that states that the radar backscatter from a forest canopy mostly originates from scattering elements that are similar in size to the wavelength being used. Sometimes this is explained through attenuation effects of a full-cover forest canopies – ie. the penetration depth is in proportion to the wavelength, so that longer wavelengths can “see” down to the larger elements. Short wavelengths only see the smaller elements in the top of the canopy.

Other times I have seen this associated with “resonance” – i.e. that the resonant scattering in the Mie region results in higher than expected returns from elements similar in size to the wavelength.

There is now a third explanation that Matthew Brolly and I have been working on and published recently in the International Journal of Remote Sensing (the full paper DOI is http://www.tandfonline.com/doi/abs/10.1080/01431161.2012.715777 for subscribers, and a here for a pre-publication proof). It does not rely on resonance nor attenuation, and so is particular relevant in areas of sparse woodland, rather than full-cover canopies (since all the elements of a tree may be visible).

A figure from the paper showing how a the backscatter response peaks at a particular branching level (not the same as height). The different lines correspond to different conditions determined by the West, Brown and Enquist model.

Our explanation is based on a combination of the trends in tree architecture and the transition from Rayleigh to optical scattering. The novelty of the approach is that we use the generic macroecological structure model introduced by West, Brown and Enquist (WBE) to explore the variability of number density with size of the scatterers. There is a finite range of ways that a tree will grow that maintains both its structural integrity and its biological function. This allows us to ask questions such as, “As branching elements get smaller, do they increase in number fast enough to compensate for their decreasing individual radar cross-section?”

For example, what if we consider only optical scattering from forest elements. The total cross-section of a volume of wood is much larger if you chop it into smaller twig-sized chunks, rather than clumping it all together in one large trunk. That is because more of the total wood volume is now exposed, rather than being obscured by the rest of the volume. Mathematically, if the length of each branch or twig is proportional to its radius, then the optical scattering goes up with the radius squared.

So, in a forest canopy, if everything scatters in an optical-type scattering kind of way, then the smaller elements at the top scatter more than the big chunky elements near the bottom (per unit volume). We can expect optical-type scattering to dominate when the scatters are large compared to the wavelength. This is encouraging – it tallies with our expectation, that shorter wavelength radar will mostly scatter from the smallest elements, not because they are near the top, but because they are sufficiently numerous.

On the other hand, when the wavelength gets much longer than the scattering elements, we are into the Rayleigh scattering regime. Here the trend is different, because the cross-section of a cylinder increases with the square of the volume of the cylinder. Using the same criteria for twigs and branches as above, this means the backscatter increases with the radius to the power of six! Clumping material together therefore increases backscatter considerably – the backscatter of the whole is greater than the sum of its parts. In a forest canopy where all the elements are small enough to be Rayleigh scatterers, the largest backscatter therefore comes from the largest elements. This also tallies with expectation; when you look at a forest with VHF radar with wavelengths a few metres long, meaning everything is a Rayleigh scatterer and the trunks are the largest contributor to the backscatter. This is exactly what is observed.

But what about the intermediate frequencies, such as L and P-band? In many circumstances we may have a forest canopy that tends to Rayleigh scattering at the top, but tends to optical scattering at the bottom. That is, the scattering elements at the top are much smaller than the wavelength, whereas at the bottom they are larger. The WBE model allows us to look at the trends since it constrains the size-to-number density relationship to within sensible limits.

Our results indicate that there is a peak response – that is, there is always one branching layer that dominates the scattering. In the extremes of very long or very short wavelengths, this dominant layer is the trunk layer, or the smallest twig layer, respectively. This concurs with results from modelling and experiment. But note that this explanation is different from the traditional explanations. It is entirely independent of attenuation through the canopy or of any resonant scattering.

The full implications of these findings we are still working on, but it’s clear that this is another example of where applying macroecological principles of tree structure and growth allows new insight into how microwaves interact with forest canopies.